Found problems: 85335
2022 IMO Shortlist, G3
Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.
2012 Traian Lălescu, 3
There are $n$ natural numbers written on a blackboard, where $n\in\mathbb{N},\ n\geq 2$. During each step two chosen numbers $a,b$, having the property that none of them divides the other, are replaced by their greatest common divisor and least common multiple. Prove that after a number of steps, all the numbers on the blackboard cease modifying. Prove that the respective number of steps is at most $(n-1)!$.
2025 Ukraine National Mathematical Olympiad, 11.6
Oleksii chose $11$ pairwise distinct positive integer numbers not exceeding $2025$. Prove that among them, it is possible to choose two numbers \(a < b\) such that the number \(b\) gives an even remainder when divided by the number \(a\).
[i]Proposed by Anton Trygub[/i]
2009 AMC 12/AHSME, 21
Let $ p(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$, where $ a$, $ b$, and $ c$ are complex numbers. Suppose that
\[ p(2009 \plus{} 9002\pi i) \equal{} p(2009) \equal{} p(9002) \equal{} 0
\]What is the number of nonreal zeros of $ x^{12} \plus{} ax^8 \plus{} bx^4 \plus{} c$?
$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$
2014 CHMMC (Fall), 4
Let $b_1 = 1$ and $ b_{n+1} = 1 + \frac{1}{n(n+1)b_1b_2...b_n}$ for $n \ge 1$. Find $b_12$.
2009 CentroAmerican, 1
Let $ P$ be the product of all non-zero digits of the positive integer $ n$. For example, $ P(4) \equal{} 4$, $ P(50) \equal{} 5$, $ P(123) \equal{} 6$, $ P(2009) \equal{} 18$.
Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).
2016 CMIMC, 3
Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$, and $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Let point $P$ be the unique point in the plane $ABC$ such that $\triangle PBM\sim\triangle PNC$. What is $AP$?
1998 Iran MO (3rd Round), 3
Let $ABC$ be a given triangle. Consider any painting of points of the plane in red and green. Show that there exist either two red points on the distance $1$, or three green points forming a triangle congruent to triangle $ABC$.
1978 Bundeswettbewerb Mathematik, 1
A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be even.
2018 Junior Balkan Team Selection Tests - Romania, 2
Let $x, y,z$ be positive real numbers satisfying $2x^2+3y^2+6z^2+12(x+y+z) =108$. Find the maximum value of $x^3y^2z$.
Alexandru Gırban
2019 German National Olympiad, 4
Show that for each non-negative integer $n$ there are unique non-negative integers $x$ and $y$ such that we have
\[n=\frac{(x+y)^2+3x+y}{2}.\]
1998 Moldova Team Selection Test, 8
Let $M=\{\frac{1}{n}|n\in\mathbb{N}\}$. Numbers $a_1,a_2,\ldots,a_l$ from an [i]arithmetic progression of maximum length[/i] $l$ $(l\geq 3)$ if they verify the properties:
a) numbers $a_1,a_2,\ldots,a_l$ from a finite arithmetic progression;
b) there is no number $b\in M$ such that numbers $b,a_1,a_2,\ldots,a_l$ or $a_1,a_2,\ldots,a_l, b$ form a finite arithmetic progression. For example numbers $\frac{1}{6},\frac{1}{3},\frac{1}{2}\in M$ form an arithmetic progression of maximum length $3$.
a) FInd an arithmetic progression of maximum length $1998$.
b) Prove that there exist maximum arithmetic progressions of any length $l \geq 3$.
2023 Moldova Team Selection Test, 11
Find all sets $ A$ of nonnegative integers with the property: if for the nonnegative intergers $m$ and $ n $ we have $m+n\in A$ then $m\cdot n\in A.$
2001 China Western Mathematical Olympiad, 2
$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value,
a) Prove that $ AB \geq 2BC$,
b) Find the value of $ AQ \cdot BQ$.
1997 Putnam, 2
$f$ be a twice differentiable real valued function satisfying
\[ f(x)+f^{\prime\prime}(x)=-xg(x)f^{\prime}(x) \]
where $g(x)\ge 0$ for all real $x$. Show that $|f(x)|$ is bounded.
1978 Romania Team Selection Test, 7
Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?
2016 Iran MO (3rd Round), 2
Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter?
( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)
2014-2015 SDML (High School), 13
Six points are chosen on the unit circle such that the product of the distances from any other point on the unit circle is at most $2$. Find the area of the hexagon with these six points as vertices.
$\text{(A) }\frac{1}{2}\qquad\text{(B) }\frac{3}{2}\qquad\text{(C) }\frac{\sqrt{3}}{2}\qquad\text{(D) }\frac{3\sqrt{3}}{2}\qquad\text{(E) }\frac{3+\sqrt{3}}{2}$
2015 BMT Spring, 16
Five points $A, B, C, D$, and $E$ in three-dimensional Euclidean space have the property that $AB = BC = CD = DE = EA = 1$ and $\angle ABC = \angle BCD =\angle CDE = \angle DEA = 90^o$ . Find all possible $\cos(\angle EAB)$.
2017 ASDAN Math Tournament, 25
Consider the sequence $\{a_n\}$ defined so that $a_n$ is the leftmost digit of $2^n$. The first few terms of this sequence are $1,2,4,8,1,3,6,\dots$. For how many $0\le n\le100000$ is $a_n=1$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max\left(0,25-\tfrac{1}{6}\sqrt{|A-C|}\right)$.
1989 Federal Competition For Advanced Students, P2, 4
We are given a circle $ k$ and nonparallel tangents $ t_1,t_2$ at points $ P_1,P_2$ on $ k$, respectively. Lines $ t_1$ and $ t_2$ meet at $ A_0$. For a point $ A_3$ on the smaller arc $ P_1 P_2,$ the tangent $ t_3$ to $ k$ at $ P_3$ meets $ t_1$ at $ A_1$ and $ t_2$ at $ A_2$. How must $ P_3$ be chosen so that the triangle $ A_0 A_1 A_2$ has maximum area?
2019 Math Prize for Girls Problems, 1
In the USA, standard letter-size paper is 8.5 inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly zero) of 8.5's and a whole number (possibly zero) of 11's?
2007 F = Ma, 31
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
Find the ratio $L/d$.
$ \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $
2007 Puerto Rico Team Selection Test, 6
The geometric mean of a set of $m$ non-negative numbers is the $m$-th root of the product of these numbers. For which positive values of $n$, is there a finite set $S_n$ of $n$ positive integers different such that the geometric mean of any subset of $S_n$ is an integer?
2015 AMC 10, 5
Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test?
$\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$